Permutations with Prescribed Pattern. II. Applications

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<ul><li><p>Math. Xachr. 83,101-126 (1978) </p><p>Permutations with Prescribed Pattern </p><p>11. Applications </p><p>By L. CARLITZ of Durham (U.S.A.) </p><p>(Eingegangen am 5.1.1976) </p><p>1. Introduction and summary. LetZM = { 1,2, . . . , n} and let 1~ = (ai, a2, . . . , a,) denote an arbitrary permutation of 2,. Let k, , k,, . . . , k, be positive integers such that </p><p>(1.1) k l + k 2 + . . .+k,=n. Thepermutation ~d will be said to have the pattern [k,, k, . . . , k,] if the following conditions are satisfied : </p><p>(1 4 al-=al</p></li><li><p>102 Carlitz, Permutations with Prescribed Pattern </p><p>For brevity put </p><p>(k i , k2, . (k l+ka+. - - + k m ) ! - 9 k,) = k l ! kz! . . . k,! * Then </p><p>m </p><p>r = i A ( k i , kq, . . , k,)= C 8, 9 (1.4) </p><p>where </p><p>(1.5) ST = 2 ( s i , ~ 2 , * * * 9 8,) and </p><p>(1.6) s , = k l + . .+kj i , sz=kji+i+- - *+kji+jz 3 . - 3 S r = k j i s ...+ j , - i+ i+* . .+ kii+ ...+ i, </p><p>and the summation in the right member of (1.5) is over all ji, . . . , j, such that (1.7) j,+j,+. . .+j,=rn, j I &gt; O , j2&gt;0 , . , . , j , &gt; O . </p><p>This result theoretically enables one to compute A(ki, k2, . . . , kr) for arbit- rary ki. However i t is rather complicated and so not really satisfactory. Thus it seems of interest to see what can be done in special case?,. A number of examples are worked out in [Z]. In the first place it is shown that if </p><p>f(n, m)= c A(k1, * - - f km) , E l + ...+ km=n </p><p>kp- 0 </p><p>(1.8) </p><p>then f (n, m) is equal to the Eulerian number An,, which enumerates tho number of nEZ, with m rises. </p><p>(1.9) </p><p>Next put </p><p>g(n, m)= A(kL, - . . , k,) . k1+ ...+ k,=n </p><p>R p l </p><p>It was proved that </p><p>where a, /3 are the roots of x%--z+y=O. The EuLERian number An,, satisfies [4] </p><p>ex - e - xyS r.s=O (r+si- i ) ! xeu-yex </p><p>m c A ( ? , 8 ) where </p><p>A (9 8) = A , +s+i,s+i =A, +s+i,r + I = A (8, ) * This suggests defining the array of numbers A(?, s) by means of </p></li><li><p>Carlite, Permutations with Prescribed Pattern 103 </p><p>It t,hen follows from (1.10) that ffl </p><p>g (n,m)= E ( - I ) ~ - ~ - - - g(2m, m) = C ( - I ) " - ~ A(% -8, s) . </p><p>A(n-s, 8) (2rnK.n) s=O </p><p>ffl </p><p>e = o </p><p>(1.11) </p><p>Finally it was proved in [2] that if A,(mk)=A,(k,k, . . . , k ) , </p><p>then x d 1 (1.12) i A k ( m k ) - - </p><p>f f l = O (mk)! E',(z) ' where </p><p>Moreover if </p><p>A, (mk+t)=A,+,(k, k , . . . , k , t ) ( t z l ) , then </p><p>( t Z 1 ) ' F,,t (4 - x7nk +t </p><p>(1.13) A , ( m k f t ) -- - =--- f f l =o (mk + t ) ! FJT) </p><p>For another proof of (1.12) and ( 1.13) see [ 11. </p><p>types. In the present paper we give some additional applications. These are of two </p><p>I. Let t be a fixed integer z2 and put </p><p>ft(n, m ) = C A(k,, h, . . - hffl) , k l +... +t, =n </p><p>kd Zt </p><p>so that ft(n, m) is the number of permutations of 2, with m inclines and the number of nodes in each incline zt; also put </p><p>- xn I?,(% y) = 1 + c -- 2 ft(n, m ) y" * </p><p>,=t n! o c t m s n We show that </p><p>where d(x, y), D(z, y) are determinants of order t defined by A(z,y)=lai- ' l ( i , j = ~ , z , . . . , t ) , </p></li><li><p>104 Carlitz, Permut$tions with Prescribed Pattern </p><p>D(x, Y) = </p><p>e 9 x . . . UI uz . . . ut a: u; ...lg . . . . . . . . . . . . &amp;' ut- l ut - 1 </p><p>t 1 '' . . . and ul, uz, . . . , ut are the ro0t.s of </p><p>z L z t - i + y = o . </p><p>For t = 2 , it is easily verified that (1.14) reduces to (1.10). </p><p>[ k j , k2, . . . , k,], where Also if fJn,m) denotes the number of permutations of 2, with pattern </p><p>k i s t , . . . , k m - l S t , kmSS ( t ~ 2 , S Z ~ ) , we evaluate the generating function </p><p>m </p><p>The results are contained in Theorems 2 and 3 below. 11. P u t </p><p>9 t h m)= z k2, * * 9 km) 7 kl+ ...+ k , =n </p><p>k i ~ 0 ( 1 ~ 1 o d t ) , k i &gt; 0 </p><p>so that ga(n, m ) denotes the number of permutations of 2, with m inclines and tho number of nodes on each incline is a multiple of t ; also put </p><p>We shall show that </p><p>( t z 2 ) , I-Y (1.15) Gt(z , y) 1 -!I%@ (1 -Y)I") </p><p>where </p><p>W e also evaluate the generating function - ,nt-j n </p><p>where g l t - j (nt - j , m ) denotes the number of permutations of Zn2-+ with pattern [k,, k2, . . . , k,], where </p><p>ki=O (modt) ( l ~ i - = m ) ; k , r -j ( m o d t ) . See Theorem 5 . </p></li><li><p>Carlitz, Perniutations with Prescribed Pattern 105 </p><p>Xote that for t = l , the generating function on the right of (1.15) reduces to </p><p>where, as above, the An,m are the EuLERian numbers. </p><p>in $3 6, 7 below. Let </p><p>We show that f(n, m ) , g(n, m ) satisfy the mixed recurrences </p><p>The case t=2 of I1 is of special interest and is examined in greater detail </p><p>f(n, m) = g h m), g(n, m ) =g2,J2n-- 1, m) . </p><p>f(n, m)'=(n-m+1)g(n,m- l )+mg(n,rn) 1 g ( n f 1, m)= (n-m+ 2) f(n, - 2) + (n+ 1) f(n, m - 1) +mf(n, m ) , by means of which the enumerants can be computed. We show also that </p><p>m </p><p>2 = O m ! g(n, n-m)= c (-l)Q&amp;) A ( 2 n + m - j - l ) , </p><p>X" - where 2 A(?%) -=see z+tan x n=O a ! </p><p>and Pm,?(n), Q,n,j(n) are polynomials in n of degree j. Finally (Theorem 8) we obtain explicit formulas for f(n, m ) and g(n, m). </p><p>2. kj z t. As above let t z 2 and put (2.1) ft(n, m)= 2 kZ, - . , km) where the summation is over all ki satisfying (2.2) k l + k 2 + . . . + k m = n ; k i z t ( i = l , 2 , . . . , m ) . We also define </p><p>(2.3) - Xn </p><p>pt't(z, Y) = 1 + c 7 c f t h m ) I" n=t ~ &lt; t m s n </p><p>In order to evaluate Ft (2, y) we apply (1.4) and (1.5). Since the number of solutions k l , k,, . . . , kj of </p><p>is equal to kl+k?+. . .+k j=s , k i z t , . . . , k i s t </p><p>s - ( t - 1 ) j - 1 ( j - 1 i t follows that </p><p>m </p></li><li><p>106 </p><p>where </p><p>Carlitz, Permutations with Prescribed Pattern </p><p>j-1 The double sum </p><p>where </p><p>It follows from (2.8) that </p><p>Now let ui, u2, . . , , ut denote the roots of (2.7) zt-zI-1 +y=O, so that </p><p>1 -z+y2=(1 -u&amp; (1 -u22) . . . (1 -utz) . If we put </p><p>where the A, are independent of z , then </p><p>Alternatively ~~ a;- I </p><p>(2.10) A.= 3 (q--J * * * ( U j - a j - { ) (cxi-uai+l). . . bj-4 - </p><p>By (2.6) and (2.8), </p></li><li><p>Carlitz, Permutations with Prescribed Pattern </p><p>80 that </p><p>(2.14) d(xi, ... , x t ) = </p><p>t </p><p>j = 1 (2.11) @:'(y)= C Aja!. Substituting from (2.11) in (2.5) we get </p><p>1 1 . . . 1 ai a2 ... at . . . . . . . . . . . . ort-2 ut-2 '2 p . . . 1 I </p><p>d (x,, . . . . xt) = c,zi + Cgx, + ... + c tx t . where cj denotes the cofactor of xi in r3(xt, . . . . xt). Then by (2.10) we have (2.15) A j = a : - ' C j d t ' , where A, is defined by (2.13). </p><p>It follows from (2.11), (2.13) and (2.15) that </p><p>Also </p><p>and </p><p>where </p><p>(2.16) </p><p>107 </p><p>t 1 ( l s k e t ) . </p><p>j = 1 j = i </p><p>. . . . . . . . . . . ... at-' I </p></li><li><p>3 08 Carlitz, Permutations with Prescribed Pattern </p><p>Hence (2.12) becomes </p><p>Substituting from (2.17) in (2.4)' we get </p><p>= r=O i ( 1 """)Y')L-&amp;. At </p><p>To justify the last step note that A, =D,(O, y) and indeed </p><p>This completes the proof of the following Theorem 1. The generating function </p><p>i s evaluated by </p><p>where D,(x, y) is defined by (2.16) and A,=Dt(O, y). Thus for example, for t = 2 , a, = u , a2=/3, (2.18) reduces to </p><p>where a , f l are the roots of 22-2 +y = O . For t = 3 , ai=a, uz=B, u3=y, we get </p></li><li><p>Carlitz. Permutations with Prescribed Pattern </p><p>1 1 1 Xi= z y 2 </p><p>x2 y' 22 </p><p>109 </p><p>1 1 1 1 1 1 . N z = ex ey ez , N 3 = x y x </p><p>x2 y' 22 ex er er </p><p>The result for t = 3 suggests that the generat,ing functions </p><p>D = x ez ey e' </p><p>z2 y2 2 2 y z </p><p>Consider the effect of removing the maximal element from 3t. If this element is not on the extreme right, n breaks into two pieces, of which the one on the left has pattern [kl, . . . , k i - l , ki- 11 and the one on the right has pattern [kj+i, . . . , km], for some j, 1 z j -=m. If however the maximal element of 3t is on the extreme right, the resulting permutation has pattern [k,, km-i, k,- 11. </p><p>Now let f,,,-,(n, m ) denote the number of permutations of 2, with pattern [k,, k?, . . . km], where </p><p>Also define (3.1) k i s t , . . . , k q n - l Z t , k , Z t - l . </p><p>f , ( O , O ) = l , f&amp; m)=O (m=-O) - We then have the following recurrence: </p><p>It follows from (3.2) that </p><p>Hence, if we put </p></li><li><p>110 Carlitz, Permutations with Prescribed Pattern </p><p>Q t - , ( x , y ) = (3.7) </p><p>we get </p><p>(3.4) </p><p>1 1 . . . 1 a1 u2 . . . ut </p><p>1 u, ... . . . . . . . . . . . . . at-a t - 2 -2 </p><p>t </p><p>-. . ea2z </p><p>Substituting in (3.4), we get </p><p>It is easily verified, using (3.5), that </p><p>Continuing this process, we remove the largest element from a permutation of Zn with pattern satisfying (3.1) and get </p><p>+ f t , &amp; 2 ( % - L m ) (t=-.S) 9 where ft,Jn, m) denotes the number of permutations of 2, with pattern [k , , 4,. . , , km] such that (3.10) k i s t , . . . , k , - , ~ t , k m S s (1 5 s - d ) . By (3.8) </p><p>w,t - 1(x, Y) (3.11) - { q - A x , Y ) P + q t - Z ( X , Y) ($=-a ax </p><p>where </p></li><li><p>Carlitz, Permutations with Prescribed Pattern </p><p>(3.17) ot,t-j(~, y)= </p><p>111 </p><p>. . . . . . . . . . . . . , . . . a:--? a;-2 ... at t - 2 a j - i ea12: j - Je=2Z . . . pests </p><p>i a2 </p><p>it follows from (3.5) and (3.11) that </p><p>Then, by (3.14), </p><p>Since (3.15) holds for j = 2 and j = 3, it follows that </p><p>(3.16) </p><p>It follows from (3.6) and (3.7) that </p><p>where 1 1 ... 1 I "i a2 ... at </p><p>(1 Sj4) . </p><p>Thus (3.16) becomes </p><p>This completes the proof of Theorem 2 . Let f,,,(n, m ) denote the number of permutations of ZN with pattern </p><p>[ki, k2, . . . , k,], where </p><p>Put ki z f, . . . , kwtk.,,_, ~ t , k, z s . </p><p>Then, for 1 Ss-= t , </p><p>where D,(z, y), Dt,Jx, y) are defined by (2.16) and (3.17). </p></li><li><p>112 Carlita, Permutations with Prescribed Pattern </p><p>1 1 ... 1 GI1 GI2 . . . Glt A , = . . . . . . . . . . . (.J-l or#--l &amp;-i </p><p>t ., . . . t </p><p>t t </p><p>(3.20) A,=C C j , D,(x, y ) = c Cieix j=1. j = j </p><p>and, by (3.17), </p><p>(3.21) Dt, t -e(z , y)=-- Cjc(eiXi (1 s s - = t ) . Thus, in place of (3.19), we may write </p><p>l t </p><p>Y ,=1 </p><p>(3.22) F J x , ~ ) = j = ~ (1 s s c t ) . </p><p>To remove the restriction s&lt; t , we note first that (3.14) is in fact valid for all 8 ~ 1 . Then, by (3.5) and (3.14) with s = t - 1 , </p><p>2-Q </p><p>It follows easily that </p><p>as might have been anticipated. Generally, by (3.14) and (3.5), </p><p>a (3.24) &amp; (DtPt,8+J=DtFt,8 ( sg l ) </p><p>We assume that </p><p>where 8 - t ,r </p><p>I n view of (3.23), (3.25) holds for s=t . </p></li><li><p>C'arlitz, Permutations wit11 Prescribed Pattern 113 </p><p>By (3.24) and (3.26), t </p><p>Q+ </p><p>DtD; , ,+ j=C Cj {Pm+l(jr)-e </p><p>ft,,+l(n,.m)=O (n-=r+l) 9 </p><p>} x:-'-' t K , ( y ) , j=i </p><p>where K J y ) is independent of x. Since </p><p>it follows that K,(y)=O. Therefore (3.25) holds for all s ~ t . </p><p>Theorem 3. With the notation of Theorem 2, we kace. for all s g t This proves the following theorein complementary to Theorem 2. </p><p>whew Cj i s the cofactor of the element in the first row and j-th column of A , and (qx) is defined by (3.26). </p><p>4. kj=O (mod t ) . Let t z - 2 and put (4-1) gt(n. ~ L ) = C kzt . . , kN2) t where the summation is over all positive kl such that (4.2) k l + k z + . . .+km=n, k j=O(modt) ( j = 1 , 2 , . . . , m ) . Thus gJn, m) is equal to the number of permutations of 2, with m inclines and the number of nodes on each incline is a multiple of t . Put </p><p>Then, by (4.1), kit +... +kmt </p><p>X - - A(kf f , . . . , k,t) - - </p><p>(kit+. . . t k , t ) ! ' G,(IL., y) = 1 + c y" m = l Ep..,km=l We now apply (1.4) and (1.5). Since the number of solutions of </p><p>k , + . . .+Tcj=s, k l ~ 1 , . . . , k j Z 1 is equal to </p><p>(;I:) ' it follows that </p><p>M sit +... +s,t </p><p>where </p><p>8 Math. Bachr. Bd. 83 </p></li><li><p>114 Carlitz, Permutations with Prescribed Pattern </p><p>r = I m=O </p><p>We have </p><p>Hence (4.3) becomes </p><p>Put </p><p>Then </p><p>so that </p><p>We may state </p><p>Theorem 4. The generating function - x"t - </p><p>G&amp;, ~ ) = 1 + ,.JJ __ C g t ( m t , m) 9" *=i (w!m=i ( t ~ 2 ) , </p><p>is evaluated by (4.5) and (4.4). The result is in fact valid for t z 1. </p></li><li><p>Cnrlitz, Permutations with Prescribed Pattern 115 </p><p>5. kj=O (mod t)-continuation. Let n denot,e a permutation with psttern [ k , , k2, . . . , kml, where kj=O (mod t ) , j = 1 ,2 , . . . , m, and consider the effect of removing the largest element of n. </p><p>lt.31. </p><p>~ </p><p>If this element is not on Dhe extreme right, the given permutation breaks into two pieces, of which the one on the left has j t - I nodes and the one on t,he right has (m -j) t nodes, for some j, 1 sj sn. </p><p>Let yt,t-I (nt- 1, m) denote the number of permutations of ZM-l with m inclines, in which the number of nodes in each incline, except the last, is divisible by t , while t.he last contains rt - 1 , for some r z 1. Also define </p><p>g1(0, 0)=1, gt(0, m)=O (m=-O) * </p><p>Then we have the recurrence </p><p>It follows from (5.1) that </p><p>Hence, if we put </p><p>we get </p></li><li><p>116 Carlitz, Permutations with Prescribed Pattern </p><p>Continuing this process, we remove the largest element from an admissible per- mutation of Zn, - and get, </p><p>1 , nt-k) </p><p>+g t , ,_ ,W-% m) (t=-2) , where gt, t Jnt - r , m), 1 57- -=t, denotes the number of permutations of Z,, --r, with m inclines, in which the number of nodes in each incline is a multiple of t , while the last contains kt - T nodes. </p><p>It follows from (5.6) that </p><p>where of course </p><p>It is convenient to put </p><p>Then (5.7) gives </p><p>60 that </p><p>Generally we have </p></li><li><p>Crtl.litz. Permutations with Prescribed Pattern 117 </p><p>This is equivalent to </p><p>The left hand side of (5.13) is equal to </p><p>p"(z) f ' (z) f ( t -1 ) f'(z"'t-'' (4 - Y __ (1 -Y) vt(x (1 - Y P t ) f(4 f(.) f b ) f ( x ) f ( 4 1 -YV&amp; (1 - 9 ) Y +- - </p><p>1-Y f (4 </p><p>--(1-y)+-. - 1-Y - - ( I - y ) + ___ 1 - y'Pt(x ( 1 - y)l't) </p><p>Thus we have verified (5.13). To sum up the results of this section we state </p><p>Theorem 6. The generating function - ,,at - j n </p><p>satisfies (5.1 1 ) for 1 sj-=t. </p><p>6. The case t = 2. We shall now examine the case t = 2 in greater detail. It will be convenient to put </p><p>F=P(x , y)=G,(x,y) G =G(x , 9 ) =G2,1(z, y) . (6.1) { </p><p>It follows that </p><p>(6.2) , -- y) 1 -y sinh (XI=) I G(z, y) = I 1 - y cosh (XI=) </p></li><li><p>118 Carlitz, Permutations with Prescribed Pattern </p><p>This is equivalent to </p><p>To facilitate the computation of partial derivatives we put </p><p>(6.7) </p><p>- y (1 -y) sinh x F, = -~ = P G , </p><p>G, = + - 1 F = - + </p><p>(1 -y Gosh x)2 1 - </p><p>_. =-1+--F+@, - y cosh x y2 sinh2 x </p><p>I -y cash x (1 -y cash x)Z 1 - Y (1 -y) cash x - P + F * - - </p><p>1 -y cosh x sinh x y sinh x cosh x FG </p><p>I - -ycoshx+ (l-ycoshx)2 y ( 1 - y ) </p><p>y sinh x </p><p>(I -y cosh x) y ( I -y) II </p><p>- 0, = - -~ </p><p>Moreover, since y cosh x </p><p>we have </p><p>This gives </p><p>1 -y - 2P + (1 + y) P2= (1 - y) G2 , It follows from these relations that </p><p>- (6.9) P z = Y (1 -9) q/ and </p><p>(6.10) ( l - y ) Q z - y ( I - Y ~ ) F g = y F . In view of (6.7), (6.9) is equivalent to </p><p>(6.11) f m = ~ (1 4 &amp;(Y) </p></li><li><p>Carlits, Permutations with Prescribed Pattern 119 </p><p>while (6.10) is equivalent to </p><p>16-12) </p><p>Hence, by ( 6 4 , (6.11) and (6.12) become (6.13) </p><p>and </p><p>(6.14) </p><p>respectively. </p><p>(1 -Y) gfls.1 -Y (1 --Y*)f;(Y)=Yf?l(Y) - </p><p>f n ( Y ) = ~ Y g , ( Y ) +Y (1 -9) 92Y) </p><p>S,+l(Y) = ( ( n + 1) Y + W 2 ) f ,(Y) +Y ( 1 -Y2) f 2 Y ) 9 </p><p>Comparing coefficients of powers of y in (6.13) and (6.14), we get </p><p>Theorem 6. The enumerants f (n, k ) , g(n, k ) satisfy the mixed recurrences (6.15) f ( n , k ) = ( n - k + l ) g ( n , k - l ) + k g ( n , k), (6.16) g ( n + l , k ) = ( n - k + 2 ) f ( n , k - 2 ) + ( n f l ) f ( n , k - l ) + k f ( n , E). </p><p>It follows from (6.15) and (6.16) that </p><p>fl </p><p>It. is evident from the definitions that (6.19) f(n, n)=A(2n), g(n, % ) = A (212-1) , where A (...</p></li></ul>